Describe how a row operation affects the determinant of a matrix.
Subsection5.1.1Warm Up
Activity5.1.1.
Consider the linear transformation \(T\colon \IR^2\to\IR^2\) corresponding to the standard matrix \(A=\left[\begin{matrix}1 & 3\\-1 & 2\end{matrix}\right]\text{.}\)
(a)
Draw a figure that depicts how \(T\) transforms the unit square.
(b)
What geometric features of the unit square were preserved by the transformation? Which geometric features changed?
Subsection5.1.2Class Activities
Activity5.1.2.
The image in Figure 46 illustrates how the linear transformation \(T : \IR^2 \rightarrow \IR^2\) given by the standard matrix \(A = \left[\begin{array}{cc} 2 & 0 \\ 0 & 3 \end{array}\right]\) transforms the unit square.
(a)
What are the lengths of \(A\vec e_1\) and \(A\vec e_2\text{?}\)
(b)
What is the area of the transformed unit square?
Activity5.1.3.
The image below illustrates how the linear transformation \(S : \IR^2 \rightarrow \IR^2\) given by the standard matrix \(B = \left[\begin{array}{cc} 2 & 3 \\ 0 & 4 \end{array}\right]\) transforms the unit square.
(a)
What are the lengths of \(B\vec e_1\) and \(B\vec e_2\text{?}\)
(b)
What is the area of the transformed unit square?
Observation5.1.4.
It is possible to find two nonparallel vectors that are scaled but not rotated by the linear map given by \(B\text{.}\)
The process for finding such vectors will be covered later in this chapter.
Observation5.1.5.
Notice that while a linear map can transform vectors in various ways, linear maps always transform parallelograms into parallelograms, and these areas are always transformed by the same factor: in the case of \(B=\left[\begin{array}{cc} 2 & 3 \\ 0 & 4 \end{array}\right]\text{,}\) this factor is \(8\text{.}\)
Since this change in area is always the same for a given linear map, it will be equal to the value of the transformed unit square (which begins with area \(1\)).
Remark5.1.6.
We will define the determinant of a square matrix \(B\text{,}\) or \(\det(B)\) for short, to be the factor by which \(B\) scales areas. In order to figure out how to compute it, we first figure out the properties it must satisfy.
Activity5.1.7.
The transformation of the unit square by the standard matrix \([\vec{e}_1\hspace{0.5em} \vec{e}_2]=\left[\begin{array}{cc}1&0\\0&1\end{array}\right]=I\) is illustrated below. If \(\det([\vec{e}_1\hspace{0.5em} \vec{e}_2])=\det(I)\) is the area of resulting parallelogram, what is the value of \(\det([\vec{e}_1\hspace{0.5em} \vec{e}_2])=\det(I)\text{?}\)
The value for \(\det([\vec{e}_1\hspace{0.5em} \vec{e}_2])=\det(I)\) is:
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4
Activity5.1.8.
The transformation of the unit square by the standard matrix \([\vec{v}\hspace{0.5em} \vec{v}]\) is illustrated below: both \(T(\vec{e}_1)=T(\vec{e}_2)=\vec{v}\text{.}\) If \(\det([\vec{v}\hspace{0.5em} \vec{v}])\) is the area of the generated parallelogram, what is the value of \(\det([\vec{v}\hspace{0.5em} \vec{v}])\text{?}\)
The value of \(\det([\vec{v}\hspace{0.5em} \vec{v}])\) is:
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4
Activity5.1.9.
Describe the value of \(\det([c\vec{v}\hspace{0.5em} \vec{w}])\text{:}\)
B. \(c\det([\vec{v}\hspace{0.5em} \vec{w}])\text{:}\) the value of \(c\) directly scales the determinant area because it scales the base of the parallelogram.
Activity5.1.10.
Describe the value of \(\det([\vec{u}+\vec{v}\hspace{0.5em} \vec{w}])\text{.}\)
The determinant is the unique function \(\det:M_{n,n}\to\IR\) satisfying these properties:
\(\displaystyle \det(I)=1\)
\(\det(A)=0\) whenever two columns of the matrix are identical.
\(\det[\cdots\hspace{0.5em}c\vec{v}\hspace{0.5em}\cdots]=
c\det[\cdots\hspace{0.5em}\vec{v}\hspace{0.5em}\cdots]\text{,}\) assuming no other columns change.
\(\det[\cdots\hspace{0.5em}\vec{v}+\vec{w}\hspace{0.5em}\cdots]=
\det[\cdots\hspace{0.5em}\vec{v}\hspace{0.5em}\cdots]+
\det[\cdots\hspace{0.5em}\vec{w}\hspace{0.5em}\cdots]\text{,}\) assuming no other columns change.
Note that these last two properties together can be phrased as “The determinant is linear in each column.”
Observation5.1.12.
The determinant must also satisfy other properties. Consider \(\det([\vec v \hspace{1em}\vec w+c \vec{v}])\) and \(\det([\vec v\hspace{1em}\vec w])\text{.}\)
The base of both parallelograms is \(\vec{v}\text{,}\) while the height has not changed, so the determinant does not change either. This can also be proven using the other properties of the determinant:
Swapping columns may be thought of as a reflection, which is represented by a negative determinant. For example, the following matrices transform the unit square into the same parallelogram, but the second matrix reflects its orientation.
To summarize, we’ve shown that the column versions of the three row-reducing operations a matrix may be used to simplify a determinant in the following way:
Multiplying a column by a scalar multiplies the determinant by that scalar:
The transformation given by the standard matrix \(A\) scales areas by \(4\text{,}\) and the transformation given by the standard matrix \(B\) scales areas by \(3\text{.}\) By what factor does the transformation given by the standard matrix \(AB\) scale areas?
\(\displaystyle 1\)
\(\displaystyle 7\)
\(\displaystyle 12\)
Cannot be determined
Fact5.1.17.
Since the transformation given by the standard matrix \(AB\) is obtained by applying the transformations given by \(A\) and \(B\text{,}\) it follows that
Find a matrix \(R\) such that \(B=RA\text{,}\) by applying the same row operation to \(I=\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}\right]\text{.}\)
(b)
Find \(\det R\) by comparing with the previous slide.
(c)
If \(C \in M_{4,4}\) is a matrix with \(\det(C)= -3\text{,}\) find
So we may compute the determinant of \(\left[\begin{array}{cc} 2 & 4 \\ 2 & 3 \end{array}\right]\) by using determinant properties to manipulate its rows/columns to reduce the matrix to \(I\text{:}\)
Suppose we have a linear transformation \(T\colon\IR^2\to\IR^2\text{.}\) Given some shape \(S\) in the plane \(\IR^2\text{,}\) we can use to \(T\) to transform it into some new shape \(T(S)\text{.}\) Consider the following questions about properties that may or may not be preserved by \(T\text{.}\)
(a)
If \(S\) is a straight line segment, explain why \(T(S)\) is also a straight line segment.
(b)
If \(S\) is a straight line segment, does \(T(S)\) necessarily have to have the same length as that of \(S\text{?}\)
(c)
If \(S\) is a triangle, explain why \(T(S)\) is also a triangle.
(d)
Continuing as above, do the angles of \(T(S)\) necessarily have to be the same as those of \(S\text{?}\)